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I recently came across the concept of a $\kappa$-cones of a metric space (Chapter I.5.2) of Bridson and Haefliger's book. In their Proposition 5.8, the provide some intuition of $\kappa$-cones by showing that $C_{\kappa}M_1^n\cong M_{\kappa}^n$ for any $\kappa\in \mathbb{R}$; where $C_{\kappa}X$ denotes the $\kappa$-cone of a metric space $X$ and $\cong$ denotes isometry.

Are there any good references focusing on $\kappa$-cones in some details, to gain some intuition on the topic? Specifically, I'd like to understand how $\kappa$-cones:

  • "functoriality" i.e. is $C_{\kappa_1}C_{\kappa_2}X\cong C_{\kappa_1+\kappa_2}X$? (or in particular, can $C_{\kappa}M_{\kappa}^n$ be identified with something simple such as some $M_{\tilde{\kappa}}$ for some $\tilde{\kappa}\in \mathbb{R}$?)
  • How does $C_{\kappa}$ interact with $p$-products of metric spaces?
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  • $\begingroup$ I think the idea is that in all three model spaces spheres around points are isometric up to rescaling to the standard sphere. One way to see this is that the three spaces are the only simply connected Riemannian manifolds, where the isometry group acts transitively on the frame bundle. But this property passes to spheres around points. Now it would be nice, if we had a way to reconstruct the balls from the spheres. As we have already seen without knowing in which of the three spaces we are, this is impossbile (the spheres are in all 3 cases isometric). $\endgroup$
    – HenrikRüping
    Commented Jul 18, 2023 at 14:18
  • $\begingroup$ But if we knew additionally the curvature $\kappa$, there is such a construction, namely $C_\kappa$. $\endgroup$
    – HenrikRüping
    Commented Jul 18, 2023 at 14:19
  • $\begingroup$ It turns out that this contruction makes sense, even if we start with more general spaces than Riemannian spheres. Usually taking $C_\kappa$ increeases the topological dimension by one, so I would not expect the first equation to hold in any sense. I dont know what a $p$-product is, so I cannot say anything about the second question. $\endgroup$
    – HenrikRüping
    Commented Jul 18, 2023 at 14:21
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    $\begingroup$ @HenrikRüping Thanks a lot for the explanation of (1). For (2) by p product I mean $d_{X\times Y}((x_1,y_1),(x_2,y_2))^p:= d_X(x_1,x_2)^p + d_Y(y_1,y_2)^p$ (for $1\le p<\infty$ and when $p=\infty$ one takes max). So $p=2$ is the usual product metric. $\endgroup$
    – Justin_other_PhD
    Commented Jul 18, 2023 at 18:58

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